Page:Grundgleichungen (Minkowski).djvu/13

If in particular the vector v{\displaystyle {\mathfrak {v}}} of the special Lorentz-transformation be equal to the velocity vector w{\displaystyle {\mathfrak {w}}} of matter at the space-time point x1,x2,x3,x4{\displaystyle x_{1},\ x_{2},\ x_{3},\ x_{4}}, then it follows out of (10), (11), (12) that

Under these circumstances therefore, the corresponding space-time point has the velocity w′=0{\displaystyle {\mathfrak {w}}'=0} after the transformation, it is as if we transform to rest. We may call the invariant ϱ1−w2{\displaystyle \varrho {\sqrt {1-{\mathfrak {w}}^{2}}}} as the rest-density of Electricity.

§ 5. Space-time Vectors. Of the 1st and 2nd kind.

If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while α14,α24,α34{\displaystyle \alpha _{14},\ \alpha _{24},\ \alpha _{34}}, are purely imaginary, but α44{\displaystyle \alpha _{44}} is real and >0{\displaystyle >0}, and x12+x22+x32+x42{\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}} transforms into x1′2+x2′2+x3′2+x4′2{\displaystyle x_{1}^{'2}+x_{2}^{'2}+x_{3}^{'2}+x_{4}^{'2}}. The operation shall be called a general Lorentz transformation.

If we put x1′=x′,x2′=y′,x3′=z′,x4′=it′{\displaystyle x'_{1}=x',\ x'_{2}=y',\ x'_{3}=z',\ x'_{4}=it'} then immediately there occurs a homogeneous linear transformation